
Covariantisation: We have looked at fermionic theories on flat Carroll backgrounds. One of the immediate things to do is to find a covariant formulation in terms of the underlying geometric data and see how fermions couple of non-flat Carroll backgrounds.

Understanding the limit: As we briefly discussed earlier, it is of importance to understand the Carrollian limit of relativistic fermions and figure out how the structures we discussed in this paper arises naturally from their relativistic parent theories. This would give more insight to the origin of the two different species of fermions we have found in the Carrollian world. Clarification as to whether the lower and upper fermions are electric and magnetic theories in the current nomenclature would also help clarify some properties of the theories we have discovered from the intrinsic analysis.

Dimensional reduction: We discussed that the lower fermions behaved like one-dimensional objects. It would be useful to quantify this further. Investigation of the spin-statistics theorem for Carrollian theories may hold important clues in this regard. It is possible that the lower fermions are like anyons in the relativistic set up. Relatedly, it would be of interest to investigate whether there is a reduction of our upper fermions. Recently it has been shown that the spacelike mass Carroll scalar theory can be reduced to a relativistic Euclidean CFT in one lower dimension [1]}. Whether a similar thing holds for the spacelike or upper fermions would be rather intriguing to work out.

More condensed matter applications: We have just begun to uncover Carroll symmetry in condensed matter systems. Even the case of twisted bi-layer graphene which we discussed in this paper needs to be understood better by looking at non-trivial boundary conditions. It would be exciting to look at the physics of e.g. quantum Hall systems. Again the possible connection to lower fermions which behave somewhat like anyons is tantalizing.

Supersymmetry: Supersymmetry in Carrollian systems, especially in conformal Carrollian ones, have been studied from the point of view of the algebra and different contractions of the parent relativistic algebras. See e.g. [2]} for a recent account of this in generic dimensions and references therein. But now, armed with the knowledge of the Carroll Clifford algebra, and the properties of Carroll fermions, it will be instructive to revisit this and construct supersymmetry in the Carroll case from scratch. This is work in progress.

